Electronic Journal of Diﬀerential Equations, Vol. 2012 ... · 2 J. C. ART´ES, J. LLIBRE, N. VULPE EJDE-2012/09 of a quadratic system. More recently the structure of this displacement

Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 09, pp. 1–35.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ftp ejde.math.txstate.edu

COMPLETE GEOMETRIC INVARIANT STUDY OF TWOCLASSES OF QUADRATIC SYSTEMS

JOAN C. ARTES, JAUME LLIBRE, NICOLAE VULPE

Abstract. In this article, using affine invariant conditions, we give a com-plete study for quadratic systems with center and for quadratic Hamiltoniansystems. There are two improvements over the results in [30] that studied cen-ters up to GL-invariant, and over the results in [1] that classified Hamiltonianquadratic systems without invariants. The geometrical affine invariant studypresented here is a crucial step toward the goal of the invariant classificationof all quadratic systems according to their singularities, finite and infinite.

1. Introduction and statement of results

Let R[x, y] be the ring of the polynomials in the variables x and y with coeffi-cients in R. We consider a system of polynomial differential equations, or simply apolynomial differential system, in R2 defined by

x = P (x, y),

y = Q(x, y),(1.1)

where P,Q ∈ R[x, y]. We say that the maximum of the degrees of the polynomialsP and Q is the degree of system (1.1). A quadratic polynomial differential systemor simply a quadratic system (QS) is a polynomial differential system of degree 2.We say that the quadratic system (1.1) is non-degenerate if the polynomials P andQ are relatively prime or coprime; i.e., g.c.d. (P,Q) = 1.

During the previous one-hundred years quadratic vector fields have been inves-tigated intensively as one of the easiest but far from trivial families of nonlineardifferential systems, and more than one thousand papers have been published aboutthese vectors fields (see for instance [24, 33, 32]). However, the problem of clas-sifying all the quadratic vector fields (even integrable ones) remains open. Formore information on the integrable differential vector fields in dimension 2, see forinstance [9, 18].

Poincare [23] defined the notion of a center for a real polynomial differentialsystem in the plane (i.e. an isolated singularity surrounded by periodic orbits).The analysis of the limit cycles which bifurcate from a focus or a center of a qua-dratic system was made by Bautin [7], by providing the structure of the powerseries development of the displacement function defined near a focus or a center

2000 Mathematics Subject Classification. 34C05, 34A34.Key words and phrases. Quadratic vector fields; weak singularities; type of singularity.c©2012 Texas State University - San Marcos.

Submitted May 16, 2011. Published January 13, 2012.

1

2 J. C. ARTES, J. LLIBRE, N. VULPE EJDE-2012/09

of a quadratic system. More recently the structure of this displacement functionhas been understood for any weak focus of a polynomial differential system. Moreprecisely, first by using a linear change of coordinates and a rescaling of the inde-pendent variable, we transform any polynomial differential system having a weakfocus or a center at the origin with eigenvalues ±ai 6= 0 (i.e. having a weak focus)into the form

x = y + P (x, y),

y = −x + Q(x, y),(1.2)

where P and Q are polynomials without constant and linear terms. Then the returnmap x 7→ h(x) is defined for |x| < R, where R is a positive number sufficientlysmall to insure that the power series expansion of h(x) at the origin is convergent.Of course, limit cycles correspond to isolated zeros of the displacement functiond(x) = h(x) − x. The structure of the power series for the displacement functionis given by the following restatement of Bautin’s fundamental result (see [25] formore details): There exists a positive integer m and a real number R > 0 suchthat the displacement function in a neighborhood of the origin for the polynomialdifferential system (1.2) can be written as

d(x) =m∑

j=1

v2j+1x2j+1

[α0 +

∞∑k=1

α2j+1k xk

],

for |x| < R, where the v2j+1’s and the α2j+1k ’s are homogeneous polynomials in the

coefficients of the polynomials P and Q.The constants Vj = v2j+1 are called the focus quantities or the Poincare-Liapunov

constants. A weak focus for which V1 = · · · = Vn−1 = 0 and Vn 6= 0 is a weak focusof order n. If all the focus quantities are zero then the weak focus is a center. Notethat any weak focus has finitely many focus quantities, in our notation exactly m.

It is known that a polynomial differential system (1.2) has a center at the origin ifand only if there exists a local analytic first integral of the form H = x2+y2+F (x, y)defined in a neighborhood of the origin, where F starts with terms of order higherthan 2. This result is due to Poincare [23] (Moussu [20] gave a geometrical proofof this result). Liapunov [17] extended Poincare result for the analytic case.

Through the coefficients of a quadratic system every one of these systems canbe identified with a single point of R12. One of the first steps in a systematicstudy of the subclasses of QS was achieved in the determining the subclass QCof all QS having a center. Of course this problem is algebraically solvable in thesense indicated by Coppel [10], because the classification of the quadratic centersis algebraically solvable.

The phase portraits of the class QC were given by Vulpe in [30] and are heredenoted by Vul # using his classification. In that classification, only GL-invariantswere used which implied that systems could only be characterized after displacingone center to the origin and adopting the standard normal form. Later papersrelated with centers provide the bifurcation diagrams for the different types ofcenters (see [26, 34, 22]).

The polynomial differential system (1.1) is Hamiltonian if there exists a poly-nomial H = H(x, y) such that P = ∂H/∂y and Q = −∂H/∂x. Regarding Hamil-tonian systems, apart from many papers using them in conservative systems, thefirst complete classification for quadratic systems was done in [1] where quadraticHamiltonian systems were split into four normal forms and a bifurcation diagram

EJDE-2012/09 COMPLETE GEOMETRIC INVARIANT 3

was provided for each one of them. No invariants were used there. Later on in[16] the affine–invariant conditions were established but they were constructed us-ing invariant polynomials of high degree without explicit geometrical meaning. Inthe later years the technique of the construction of invariant polynomials has beengreatly improved. Now, with these better tools, the invariants needed are of lowerdegree and consistent with the set of all invariants needed to describe singularpoints.

The main results of this article are the following two theorems.

Theorem 1.1. Consider a quadratic system of differential equations.(i) This system possesses a center and the configuration of all its singularities

(finite and infinite) up to a congruent equivalence, given in Table 2 if andonly if the corresponding affine invariant conditions described in Table 2hold.

(ii) We have a total of 41 congruently distinct configurations of singularities.For each phase portrait of quadratic systems with a center we have thefollowing two possibilities:(a) it corresponds to a unique configuration of singularities; there are 17

such phase portraits;(b) it corresponds to several configurations of singularities; there are 14

such phase portraits. The richest example is V ul2 which could occurwith anyone of the 8 congruently distinct configurations 15, 17, 19, 20,26, 37, 39, 41.

(iii) The phase portrait of a system with a center corresponds to the one of31 topologically distinct phase portraits constructed in [30] (except the caseof a linear system) and either it is determined univocally by the respec-tive configuration, or it is determined by the configuration and additionalconditions given in Table 3. More exactly we have 35 congruently distinctconfigurations, each of which leads univocally to a unique phase portrait;and there are 6 configurations each of which leads to several phase portraitsdistinguished by the additional conditions according to Table 3.

Tables 2 and 3 can be found in section 6.

Theorem 1.2. Assume that a quadratic system of differential equations is Hamil-tonian.

(i) This system possesses the configuration of all its singularities (finite andinfinite) up to a congruent equivalence, given in Table 4 if and only if thecorresponding affine invariant conditions described in Table 4 hold.

(ii) We have a total of 30 congruently distinct configurations of singularities.For each phase portrait of Hamiltonian quadratic systems we have the fol-lowing two possibilities:(a) it corresponds to a unique configuration of singularities; there are 23

such phase portraits;(b) it corresponds to several configurations of singularities; there are 5 such

phase portraits. The richest example is Ham11 which could occur withanyone of the 5 congruently distinct configurations 7,11, 20,26,28.

(iii) The phase portrait of this system corresponds to the one of 28 topologicallydistinct phase portraits constructed in [1] and either it is determined univo-cally by the respective configuration, or it is determined by the configuration


and additional conditions given in Table 5. More exactly we have 24 con-gruently distinct configurations, each of which leads univocally to a uniquephase portrait; and there are 6 configurations each of which leads to severalphase portraits distinguished by the additional conditions according to Table5.

Tables 4 and 5 can be found in section 7.The work is organized as follows. In sections 2 and 3 we introduce the notation

that we use for describing the singular points. In section 4 we give some preliminaryresults needed for the work. In section 5 we adapt a diagram from a previous paper[5] to describe more easily the bifurcation tree of finite singularities, and we alsointroduce the used invariants from [30] and [16]. Finally in sections 6 and 7 weprove the main theorems of this paper.

2. Equivalence relations for singularities of planar polynomialvector fields

We first recall the topological equivalence relation as it is used in most of theliterature. Two singularities p1 and p2 are topologically equivalent if there existopen neighborhoods N1 and N2 of these points and a homeomorphism Ψ : N1 → N2

carrying orbits to orbits and preserving orientation. To reduce the number of cases,by topological equivalence we shall mean here that the homeomorphism Ψ preservesor reverses the orientation. In this article we use this second notion, which issometimes used in the literature (see [12, 2]).

Polynomial vector fields can be compactified using different techniques whichgive a global view of the phase portraits including the trajectories close to infinitywhich leads to the notion of infinite singular points (for more details see, for example[19]).

Finite and infinite singular points may either be real or complex. Most of thetimes one only needs to observe the real ones. We point out that the sum of themultiplicities of all singular points of a quadratic system (with a finite number ofsingular points) is always 7. The sum of the multiplicities of the infinite singu-lar points is always at least 3, more precisely it is always 3 plus the sum of themultiplicities of the finite points which have gone to infinity.

We use here the following terminology for singularities grouped in the families:• We call elemental a singular point with its both eigenvalues not zero.• We call semi-elemental a singular point with exactly one of its eigenvalues

equal to zero.• We call nilpotent a singular point with its eigenvalues zero but its Jacobian

matrix is not identically zero.• We call intricate a singular point with its Jacobian matrix identically zero.

This notation (except “nilpotent”) was proposed by Dana Schlomiukin a personal communication in order to avoid intersection with previouswell-known notations. We are grateful to her for the help.

We say that two points are Jordan relatives if they both belong to one of thefamilies above.

Roughly speaking a singular point p of an analytic differential system χ is amultiple singularity of multiplicity m if p produces m singularities, as closed to pas we wish, in analytic perturbations χε of this system and m is the maximum


such number. In polynomial differential systems of fixed degree n we have severalpossibilities for obtaining multiple singularities.

(i) A finite singular point can split into several finite singularities in n-degreepolynomial perturbations.

(ii) An infinite singular point could split into some finite and some infinitesingularities in n-degree polynomial perturbations.

(iii) n-degree perturbations of an infinite singularity produce only infinite sin-gular points of the systems.

To all these cases we can give a precise mathematical meaning using the notion ofintersection multiplicity at a point p of two algebraic curves.

Two foci (or saddles) are order equivalent if their corresponding orders coincide.Semi-elemental saddle-nodes are always topologically equivalent.

Definition 2.1. Two singularities p1 and p2 of two polynomial vector fields arecongruently equivalent if and only if they are topologically equivalent, they havethe same multiplicity, they are Jordan relatives and in case of foci or saddles theyare order equivalent.

In this work we discuss the behavior of quadratic vector fields globally aroundtheir singularities.

Definition 2.2. Let χ1 and χ2 be two polynomial vector fields each having afinite number of singularities. We say that χ1 and χ2 have congruent equivalentconfigurations of singularities if and only if we have a bijection ϑ carrying thesingularities of χ1 to singularities of χ2 and for every singularity p of χ1, ϑ(p) iscongruently equivalent with p.

3. The notation for singular points

In this section we present the notation that we use for describing the singularpoints. The complete notation for singular points will appear in our project ofclassification of finite and infinite singular points of all QS.

This notation used here for describing finite and infinite singular points of qua-dratic systems, can easily be extended to general polynomial systems.

We start by distinguishing the finite and infinite singularities denoting the firstones with lower case letters and the second with capital letters. When describing ina row both finite and infinite singular points, we will always order them first finite,latter infinite with a semicolon (‘;’) separating them.

Starting with elemental points, we use the letters ‘s’,‘S’ for “saddles”; ‘n’, ‘N ’for “nodes”; ‘f ’ for “foci” and ‘c’ for “centers”.

An elemental singular point is called a weak singularity if the trace of its Jacobianis zero. It follows easily that such a singular point could be either a focus or a centeror a saddle. In order to determine the stability of weak focus one needs to computehigher order terms of a certain function (see [17]). Depending on the number ofthe terms of this function which vanish we can determine the order of the focus. Asimilar technique can be used also in the case of a weak saddle.

Finite elemental foci (or saddles) are classified according to their order as weakfoci (or saddles). When the trace of the Jacobian matrix evaluated at those singularpoints is not zero, we call them strong saddles and strong foci and we maintain thestandard notations ‘s’ and ‘f .’ But when the trace is zero, it is known that forquadratic systems they may have up to 3 orders plus an integrable one, which


corresponds to infinite order. So, from the order 1 to order 3 we denote them by‘s(i)’ and ‘f (i)’ where i = 1, 2, 3 is the order. For the integrable case, the saddleremains a topological saddle and it will be denoted by ‘$’. In the second case wehave a change in the topology of the local phase portrait which makes the singularpoint a center and it is denoted by ‘c’.

Foci and centers cannot appear as isolated singular points at infinity and henceit is not necessary to introduce their order in this case. In case of saddles, we canhave weak saddles at infinity but it is premature at this stage to describe themsince the maximum order of weak singularities in cubic systems is not yet known.

All non–elemental singular points are multiple points, in the sense that whenwe perturb them within a nearby system they could split in at least two elementalpoints. For finite singular points we denote with a subindex their multiplicity as in‘s(5)’ or in ‘es(3)’ (the meaning of the ‘ ’ and the ‘ ’ will be explained below). Themultiplicity of a singularity of a QS is the maximum number of singular pointswhich can appear from this singularity when we perturb it inside the class of allQS. In order to describe the various kinds of multiplicity of infinite singular pointswe use the concepts and notations introduced in [28]. Thus we denote by ‘

(ab

)...’

the maximum number a (respectively b) of finite (respectively infinite) singularitieswhich can be obtained by perturbation of the multiple point. For example ‘

(11

)SN ’

means a saddle–node at infinity produced by the collision of one finite singularitywith an infinite one; ‘

(03

)S’ means a saddle produced by the collision of 3 infinite

singularities.Semi–elemental points can either be nodes, saddles or saddle–nodes, finite or

infinite. We will denote them always with an overline, for example ‘sn’, ‘s’ and ‘n’with the corresponding multiplicity. In the case of infinite points we will put the‘ ’ on top of the parenthesis of multiplicity.

Nilpotent points can either be saddles, nodes, saddle–nodes, elliptic-saddles,cusps, foci or centers. The first four of these could be at infinity. We denotethe finite ones with a hat ‘’ as in es(3) for a finite nilpotent elliptic-saddle of mul-tiplicity 3, and cp(2) for a finite nilpotent cusp point of multiplicity 2. In the caseof nilpotent infinite points, analogously to the case of semi-elemental points we willput the ‘’ on top of the parenthesis of multiplicity. The relative position of thesectors of an infinite nilpotent point with respect to the line at infinity can producetopologically different phase portraits. This forces us to use a notation for thesepoints similar to the notation which we will use for the intricate points.

It is known that the neighborhood of any singular point of a polynomial vectorfield (except foci and centers) is formed by a finite number of sectors which couldonly be of three types: parabolic, hyperbolic and elliptic (see [11]). Then a rea-sonable way to describe intricate points and nilpotent points at infinity is to use asequence formed by the types of their sectors. The description we give is the onewhich appears in the clock–wise direction once the blow–down is done. Thus inquadratic systems we have just seven possibilities for finite intricate singular points(see [3]) which are the following ones

• (a) phpphp(4);• (b) phph(4);• (c) hh(4);• (d) hhhhhh(4);• (e) peppep(4);


• (f) pepe(4);• (g) ee(4).

We use lower case because of the finite nature of the singularities and add thesubindex (4) since they are of multiplicity 4.

For infinite intricate and nilpotent singular points, we insert a hyphen betweenthe sectors to split those which appear on one side of the equator of the spherefrom the ones which appear in the other side. In this way we distinguish between(22

)PHP -PHP and

(22

)PPH-PPH.

The lack of finite singular points will be encapsulated in the notation ∅. In thecases we need to point out the lack of an infinite singular point we will use thesymbol ∅.

Finally there is also the possibility that we have an infinite number of finite orinfinite singular points. In the first case, this means that the polynomials definingthe differential system are not coprime. Their common factor may produce a realline or conic filled up with singular points, or a conic with real coefficients havingonly complex points.

We consider now systems which have the set of non isolated singularites locatedon the line at infinity. It is known that the neighborhood of infinity can be of 6different types (see [28]) up to topological equivalence. The way to determine themcomes from a study of the reduced system on the infinite local charts where theline of singularities can be removed within the chart and still a singular point mayremain on the line at infinity. Thus, depending of the nature of this point, thebehavior of the singularities at infinity of the original system can be denoted as[∞, ∅], [∞, N ], [∞, S], [∞, C], [∞, SN ] or [∞, ES]. In the families showed in thispaper we will only meet the case [∞, S].

We will denote with the symbol the case when the polynomials defining thesystem have a common factor. The symbol stands for the most generic of thesecases which corresponds to a real line filled up of singular points. The degeneracycan be also be produced by a common quadratic factor which could generate anykind of conic. We will indicate each case by the following symbols

• [|] for a real straight line;• [∪] for a real parabola;• [‖] for two real parallel lines;• [‖c] for two complex parallel lines;• [|2] for a double real straight line;• [ )( ] for a real hyperbola;• [×] for two intersecting real straight lines;• [◦] for a real circle or ellipse;• [ c©] for a complex conic;• [·] for two complex straight lines which intersect at a real finite point.

The cases that will be considered in this paper are a subset of the previous cases.Moreover we also want to determine whether after removing the common factor

of the polynomials, singular points remain on the curve defined by this commonfactor. If the reduced system has no finite singularity which remains on the curvedefined by this common factor, we will use the symbol ∅to describe this situation. Ifsome singular points remain we will use the corresponding notation of their types.

The existence of a common factor of the polynomials defining the differentialsystem also affects the infinite singular points. We point out that the projective


completion of a real affine line filled up with singular points has a point on the lineat infinity which will then be also a non isolated singularity.

In order to describe correctly the singularities at infinity, we must mention alsothis kind of phenomena and describe what happens to such points at infinity afterthe removal of the common factor. To show the existence of the common factor wewill use the same symbol as before: , and for the type of degeneracy we use thesymbols introduced above. We will use the symbol ∅ to denote the non-existenceof infinite singular points after the removal of the degeneracy. There are otherpossibilities for a polynomial system, but this is the only one of interest in thispaper.

4. Some preliminary results

Consider real quadratic systems of the formdx

dt= p0 + p1(x, y) + p2(x, y) ≡ P (x, y),

dy

dt= q0 + q1(x, y) + q2(x, y) ≡ Q(x, y),

(4.1)

with homogeneous polynomials pi and qi (i = 0, 1, 2) of degree i in x, y, where

p0 = a00, p1(x, y) = a10x + a01y, p2(x, y) = a20x2 + 2a11xy + a02y

2,

q0 = b00, q1(x, y) = b10x + b01y, q2(x, y) = b20x2 + 2b11xy + b02y

2.

Let a = (a00, a10, a01, a20, a11, a02, b00, b10, b01, b20, b11, b02) be the 12-tuple of thecoefficients of systems (4.1) and denote R[a, x, y] = R[a00, . . . , b02, x, y].

4.1. Number and types of weak singularities of quadratic systems. A com-plete characterization of the finite weak singularities of quadratic systems via in-variant theory was done in [31], where the next result is proved.

Proposition 4.1. Consider a non-degenerate quadratic system (4.1).(a) If T4 6= 0 then this system has no weak singularity.(b) If T4 = 0 and T3 6= 0 then the system has exactly one weak singularity.

Moreover this singularity is either a weak focus (respectively a weak saddle)of the indicated order below, or a center (respectively an integrable saddle) ifand only if T3F < 0 (respectively T3F > 0) and the following correspondingcondition holds(b1) f (1) (respectively s(1)) ⇔ F1 6= 0;(b2) f (2) (respectively s(2)) ⇔ F1 = 0, F2 6= 0;(b3) f (3) (respectively s(3)) ⇔ F1 = F2 = 0, F3F4 6= 0;(b4) c (respectively $) ⇔ F1 = F2 = F3F4 = 0.

(c) If T4 = T3 = 0 and T2 6= 0, then the system could possess two and onlytwo weak singularities and none of them is of order 2 or 3. Moreover thissystem possesses two weak singularities, which are of the types indicatedbelow, if and only if F = 0 and one of the following conditions holds(c1) s(1), s(1) ⇔ F1 6= 0, T2 < 0, B ≤ 0, H > 0;(c2) s(1), f (1) ⇔ F1 6= 0, T2 > 0, B < 0;(c3) f (1), f (1) ⇔ F1 6= 0, T2 < 0, B < 0, H < 0;(c4) $, $ ⇔ F1 = 0, T2 < 0, B < 0, H > 0;(c5) $, c ⇔ F1 = 0, T2 > 0, B < 0;


(c6) c, c ⇔ F1 = 0, T2 < 0, B < 0, H < 0.(d) If T4 = T3 = T2 = 0 and T1 6= 0, then the system could possess one and

only one weak singularity (which is of order 1). Moreover this system hasone weak singularity of the type indicated below if and only if F = 0 andone of the following conditions holds(d1) s(1) ⇔ F1 6= 0, B < 0, H > 0;(d2) f (1) ⇔ F1 6= 0, B < 0, H < 0.

(e) If T4 = T3 = T2 = T1 = 0 and σ(a, x, y) 6= 0, then the system could possessone and only one weak singularity. Moreover this system has one weaksingularity, which is of the type indicated below, if and only if one of thefollowing conditions holds(e1) s(1) ⇔ F1 6= 0, H = B1 = 0, B2 > 0;(e2) f (1) ⇔ F1 6= 0, H = B1 = 0, B2 < 0;(e3)

$ ⇔

[α] F1 = 0, F = 0, B < 0, H > 0, or[β] F1 = 0, H = B1 = 0, B2 > 0, or[γ] F1 = 0, H = B = B1 = B2 = B3 = µ0 = 0, K(µ2

2 + µ23) 6= 0, or

[δ] F1 = 0, H = B = B1 = B2 = B3 = K = 0, µ2G 6= 0, or[ε] F1 = 0, H = B = B1 = B2 = B3 = B4 = K = µ2 = 0, µ3 6= 0;

(e4)

c ⇔

{[α] F1 = 0, F = 0, B < 0, H < 0, or[β] F1 = 0, H = B1 = 0, B2 < 0.

(f) If σ(a, x, y) = 0, then the system is Hamiltonian and it possesses i (with1 ≤ i ≤ 4) weak singular points of the types indicated below if and only ifone of the following conditions holds(f1) $, $, $, c ⇔ µ0 < 0, D < 0, R > 0, S > 0;(f2) $, $, c, c ⇔ µ0 > 0, D < 0, R > 0, S > 0;(f3) $, $, c ⇔ µ0 = 0, D < 0, R 6= 0;(f4)

$, $ ⇔

[α] µ0 < 0, D > 0, or[β] µ0 < 0, D = 0, T < 0, or[γ] µ0 = R = 0, P 6= 0, U > 0, K 6= 0;

(f5)

$, c ⇔

[α] µ0 > 0, D > 0, or[β] µ0 > 0, D = 0, T < 0, or[γ] µ0 = R = 0, P 6= 0, U > 0, K = 0;

(f6)

$ ⇔

[α] µ0 < 0, D = T = P = 0, R 6= 0, or[β] µ0 = 0, D > 0, R 6= 0, or[γ] µ0 = 0, D = 0, PR 6= 0, or[δ] µ0 = R = P = 0, U 6= 0;

(f7) c ⇔ µ0 > 0, D = T = P = 0, R 6= 0.


The invariant polynomials used in the above theorem are constructed as follows

F1(a) = A2,

F2(a) = −2A21A3 + 2A5(5A8 + 3A9) + A3(A8 − 3A10 + 3A11 + A12)

−A4(10A8 − 3A9 + 5A10 + 5A11 + 5A12),

F3(a) = −10A21A3 + 2A5(A8 −A9)−A4(2A8 + A9 + A10 + A11 + A12)

+ A3(5A8 + A10 −A11 + 5A12),

F4(a) = 20A21A2 −A2(7A8 − 4A9 + A10 + A11 + 7A12) + A1(6A14 − 22A15)

− 4A33 + 4A34,

F(a) = A7, B(a) = −(3A8 + 2A9 + A10 + A11 + A12),

H(a) = −(A4 + 2A5), G(a, x, y) = M + 32H,

and

B1(a) ={(

T7, D2

)(1)[12D1T3 + 2D31 + 9D1T4 + 36

(T1, D2

)(1)]− 2D1

(T6, D2

)(1)

×[D2

1 + 12T3] + D21

[D1

(T8, C1

)(2) + 6((

T6, C1

)(1), D2

)(1)]}/144,

B2(a) ={(

T7, D2

)(1)[8T3

(T6, D2

)(1) −D21

(T8, C1

)(2) − 4D1

((T6, C1

)(1), D2

)(1)]+

[(T7, D2

)(1)]2

(8T3 − 3T4 + 2D21)

}/384,

B3(a, x, y) = −D21(4D2

2 + T8 + 4T9) + 3D1D2(T6 + 4T7)− 24T3(D22 − T9),

B4(a, x, y) = D1(T5 + 2D2C1)− 3C2(D21 + 2T3).

Here by (f, g)(k) is denoted the differential operator called transvectant of index k(see [21]) of two polynomials f, g ∈ R[a, x, y]

(f, g)(k) =k∑

h=0

(−1)h

(k

h

)∂kf

∂xk−h∂yh

∂kg

∂xh∂yk−h,

and Ai(a) are the elements of the minimal polynomial basis of affine invariants upto degree 12 (containing 42 elements) constructed in [8]. We have applied here onlythe following elements (keeping the notation from [8])

A1 = A, A2 = (C2, D)(3)/12,

A3 =((

(C2, D2)(1), D2

)(1), D2

)(1)/48, A4 = (H, H)(2),

A5 = (H, K)(2)/2, A6 = (E, H)(2)/2,

A7 =((C2, E)(2), D2

)(1)/8, A8 =

((D, H)(2), D2

)(1)/8,

A9 =((

(D,D2)(1), D2

)(1), D2

)(1)/48, A10 =

((D, K)(2), D2

)(1)/8,

A11 = (F , K)(2)/4, A12 = (F , H)(2)/4,

A14 = (B, C2)(3)/36, A15 = (E, F )(2)/4,

A33 =(((

(D,D2)(1), F)(1)

, D2

)(1), D2

)(1)/128,

A34 =(((

(D, D)(2), D2

)(1), K

)(1), D2

)(1)/64,


where

A =(C1, T8 − 2T9 + D2

2

)(2)/144,

D =[2C0(T8 − 8T9 − 2D2

2) + C1(6T7 − T6 − (C1, T5)(1) + 6D1(C1D2 − T5)

− 9D21C2

]/36,

E =[D1(2T9 − T8)− 3 (C1, T9)

(1) −D2(3T7 + D1D2)]/72,

F =[6D2

1(D22 − 4T9) + 4D1D2(T6 + 6T7) + 48C0 (D2, T9)

(1) − 9D22T4 + 288D1E

− 24(C2, D)(2) + 120(D2, D)(1) − 36C1 (D2, T7)(1) + 8D1 (D2, T5)

(1)]/144,

B ={

16D1 (D2, T8)(1) (3C1D1 − 2C0D2 + 4T2) + 32C0 (D2, T9)

(1) (3D1D2

− 5T6 + 9T7

)+ 2 (D2, T9)

(1) (27C1T4 − 18C1D

21 − 32D1T2 + 32 (C0, T5)

(1) )+ 6 (D2, T7)

(1) [8C0(T8 − 12T9)− 12C1(D1D2 + T7) + D1(26C2D1 + 32T5)

+ C2(9T4 + 96T3)] + 6 (D2, T6)(1) [32C0T9 − C1(12T7 + 52D1D2)− 32C2D

21]

+ 48D2 (D2, T1)(1) (

2D22 − T8

)− 32D1T8 (D2, T2)

(1) + 9D22T4 (T6 − 2T7)

− 16D1 (C2, T8)(1) (

D21 + 4T3

)+ 12D1 (C1, T8)

(2) (C1D2 − 2C2D1)

+ 6D1D2T4

(T8 − 7D2

2 − 42T9

)+ 12D1 (C1, T8)

(1) (T7 + 2D1D2)

+ 96D22

[D1 (C1, T6)

(1) + D2 (C0, T6)(1)

]− 16D1D2T3

(2D2

2 + 3T8

)− 4D3

1D2

(D2

2 + 3T8 + 6T9

)+ 6D2

1D22 (7T6 + 2T7)− 252D1D2T4T9

}/(2833),

K = (T8 + 4T9 + 4D22)/72 ≡

(p2(x, y), q2(x, y)

)(1)

/4,

H = (−T8 + 8T9 + 2D22)/72,

M = (C2, C2)(2) = 2Hess(C2(x, y)

)and

T1 = (C0, C1)(1)

, T2 = (C0, C2)(1)

, T3 = (C0, D2)(1)

,

T4 = (C1, C1)(2)

, T5 = (C1, C2)(1)

, T6 = (C1, C2)(2)

,

T7 = (C1, D2)(1)

, T8 = (C2, C2)(2)

, T9 = (C2, D2)(1)

(4.2)

are the GL-comitants constructed by using the following five polynomials, basicingredients in constructing invariant polynomials for systems (4.1)

Ci(a, x, y) = ypi(x, y)− xqi(x, y), (i = 0, 1, 2),

Di(a, x, y) =∂pi

∂x+

∂qi

∂y, (i = 1, 2).

(4.3)

The affine invariants Tj (j = 1, 2, 3, 4) which are responsible for the number ofvanishing traces of the finite singularities (see [31]) are constructed as follows.

We consider the polynomial σ(a, x, y) which is an affine comitant of systems (4.1)

σ(a, x, y) =∂P

∂x+

∂Q

∂y= σ0(a) + σ1(a, x, y) (≡ D1(a) + D2(a, x, y)),


and the differential operator L = x ·L2 − y ·L1 (see [5]) acting on R[a, x, y], where

L1 = 2a00∂

∂a10+ a10

∂

∂a20+

12a01

∂

∂a11+ 2b00

∂

∂b10+ b10

∂

∂b20+

12b01

∂

∂b11;

L2 = 2a00∂

∂a01+ a01

∂

∂a02+

12a10

∂

∂a11+ 2b00

∂

∂b01+ b01

∂

∂b02+

12b10

∂

∂b11.

Applying the differential operators L and (∗, ∗)(k) (i.e. transvectant of index k) in[31] is defined the following polynomial function (named trace function)

T(w) =4∑

i=0

1(i!)2

(σi

1,1i!L(i)(µ0)

)(i)

w4−i =4∑

i=0

Gi w4−i, (4.4)

where the coefficients Gi(a) = 1(i!)2 (σi

1, µi)(i), i = 0, 1, 2, 3, 4(G0(a) ≡ µ0(a)

)are

GL-invariants.Finally using the function T(w) the following four needed affine invariants T4,

T3, T2, T1 are constructed [31]

T4−i(a) =1i!

diT

dwi

∣∣∣w=σ0

, i = 0, 1, 2, 3(T4 ≡ T(σ0)

),

which are basic schematic affine invariants for the characterization of weak singu-larities via invariant polynomials (see Proposition 4.1).

The invariant polynomials D,P,R,S,T,U and V are defined in the next section(see (5.2)).

In what follows we also need the following invariant polynomials:

B3(a, x, y) = (C2, D)(1) = Jacob(C2, D

),

B2(a, x, y) = (B3, B3)(2) − 6B3(C2, D)(3),

B1(a) = Resx

(C2, D

)/y9 = −2−93−8 (B2, B3)

(4),

B4(a, x, y) = −((D, H)(2), H

)(1) × (D, H)(2),

B5(a, x, y) = D2

[((C2, D2)(1), D2)(1) − 3(C2, K)(2)

],

B6(a, x, y) = C21 − 4C0C2.

(4.5)

andη(a) = Discrim

(C2(x, y)

)= (M, M)(2)/384,

N(a, x, y) = 4K − 4H; R(a, x, y) = L + 32K,

θ(a) = Discrim(N(a, x, y)

)= −(N , N)(2)/2,

L(a, x, y) = 16K − 32H − M ; θ1(a) = 16η − 2θ − 16µ0.

(4.6)

4.2. Number and multiplicities of the finite singularities of quadratic sys-tems. The conditions for the number and multiplicities of the finite singularitiesof quadratic systems were first constructed in [5].

We shall use here the notion of zero–cycle in order to describe the numberand multiplicity of singular points of a quadratic system. This notion as wellas the notion of divisor, were used for classification purposes of planar quadraticdifferential systems by Pal and Schlomiuk [27], Llibre and Schlomiuk [19], Schlomiukand Vulpe [28] and by Artes and Llibre and Schlomiuk [2].


Definition 4.2. We consider formal expressions D =∑

n(w)w where n(w) is aninteger and only a finite number of n(w) are nonzero. Such an expression is calleda zero–cycle of P2(C) if all w appearing in D are points of P2(C). We call degreeof the zero–cycle D the integer deg(D) =

∑n(w). We call support of D the set

supp(D) of w’s appearing in D such that n(w) 6= 0.

We note that P2(C) denotes the complex projective space of dimension 2. Fora system (S) belonging to the family (4.1) we denote ν(P,Q) = {w ∈ C2 : P (w) =Q(w) = 0} and we define the zero-cycle D

S(P,Q) =

∑w∈ν(P,Q) Iw(P,Q)w, where

Iw(P,Q) is the intersection number or multiplicity of intersection of P and Q atw. It is clear that for a non–degenerate quadratic system deg(D

S) ≤ 4 as well as

supp(DS) ≤ 4. For a degenerate system the zero–cycle D

S(P,Q) is undefined.

Using the affine invariant polynomials

µ0(a), D(a), R(a, x, y), S(a, x, y), T(a, x, y), U(a, x, y), V(a, x, y)(4.7)

(the construction of these polynomials will be discussed further), in [5] the nextproposition was proved.

Proposition 4.3 ([5]). The form of the divisor DS(P,Q) for non-degenerate qua-

dratic systems (4.1) is determined by the corresponding conditions indicated in Table3, where we write p + q + rc + sc if two of the finite points, i.e. rc, sc, are complexbut not real.

Table 1.

No.Zero–cycleD

S(P, Q)

Invariantcriteria

No.Zero–cycleD

S(P, Q)

Invariantcriteria

1 p + q + r + sµ0 6= 0, D < 0,R > 0, S > 0

10 p + q + r µ0 = 0, D < 0, R 6= 0

2 p + q + rc + sc µ0 6= 0, D > 0 11 p + qc + rc µ0 = 0, D > 0, R 6= 0

3 p c + qc + rc + scµ0 6= 0, D < 0, R ≤ 0

12 2p + q µ0 = D = 0, PR 6= 0µ0 6= 0, D < 0, S ≤ 04 2p + q + r µ0 6= 0, D = 0, T < 0 13 3p µ0 = D = P = 0, R 6= 0

5 2p + qc + rc µ0 6= 0, D = 0, T > 0 14 p + qµ0 = R = 0, P 6= 0,

U > 0

6 2p + 2qµ0 6= 0, D = T = 0,

PR > 015 p c + qc µ0 = R = 0, P 6= 0,

U < 0

7 2p c + 2qc µ0 6= 0, D = T = 0,PR < 0

16 2pµ0 = R = 0, P 6= 0,

U = 0

8 3p + qµ0 6= 0, D = T = 0,

P = 0, R 6= 017 p

µ0 = R = P = 0,U 6= 0

9 4pµ0 6= 0, D = T = 0,

P = R = 018 0

µ0 = R = P = 0,U = 0, V 6= 0

5. The global diagram for the finite singularities of quadraticsystems. Some needed invariants

We note that the polynomials (4.7) were constructed in [5] (see also [3]) usingthe basic ingredients (4.3) in constructing invariant polynomials for systems (4.1)and applying the differential operator (∗, ∗)(k) (i.e. transvectant of index k).

Here we shall use the new expressions for the polynomials (4.7) (constructed in[31]), which are equivalent to the old ones but make more transparent their geometry


and allow us to observe the dynamic of the finite singularities. More exactly weshall use the polynomials µ0(a) and µi(a, x, y) constructed in [5] as follows

µ0(a) = Resx

(p2(x, y), q2(x, y)

)/y4,

µi(a, x, y) =1i!L(i)(µ0), i = 1, . . . , 4,

(5.1)

where L(i)(µ0) = L(L(i−1)(µ0)). Their geometrical meaning is revealed in thefollowing two lemmas.

Lemma 5.1 ([5]). The total multiplicity of all finite singularities of a quadratic sys-tem (4.1) equals k if and only if for every i ∈ {0, 1, . . . , k−1} we have µi(a, x, y) = 0in R[x, y] and µk(a, x, y) 6= 0. Moreover a system (4.1) is degenerate if and only ifµi(a, x, y) = 0 in R[x, y] for every i = 0, 1, 2, 3, 4.

Lemma 5.2 ([6]). The point M0(0, 0) is a singular point of multiplicity k (1 ≤ k ≤4) for a quadratic system (4.1) if and only if for every i ∈ {0, 1, . . . , k− 1} we haveµ4−i(a, x, y) = 0 in R[x, y] and µ4−k(a, x, y) 6= 0.

Using the invariant polynomials µi (i = 0, 1, . . . , 4) in [31] the polynomials (4.7)are constructed as follows

D =[3((µ3, µ3)(2), µ2

)(2) −(6µ0µ4 − 3µ1µ3 + µ2

2, µ4)(4)]/48,

P = 12µ0µ4 − 3µ1µ3 + µ22,

R = 3µ21 − 8µ0µ2,

S = R2 − 16µ20P,

T = 18µ20(3µ2

3 − 8µ2µ4) + 2µ0(2µ32 − 9µ1µ2µ3 + 27µ2

1µ4)−PR,

U = µ23 − 4µ2µ4,

V = µ4.

(5.2)

Considering these expressions we have the next remark.

Remark 5.3. If µ0 = 0 then the condition R = 0 (respectively R = P = 0;R = P = U = 0; R = P = U = V = 0) is equivalent to µ1 = 0 (respectivelyµ1 = µ2 = 0; µ1 = µ2 = µ3 = 0; µ1 = µ2 = µ3 = µ4 = 0).

On the other hand, considering Lemma 5.1 we deduce that the invariant polyno-mials µi (i = 0, 1, . . . , 4) are responsible for the number of finite singularities whichhave coalesced with infinite ones. So taking into account the remark above andProposition 4.3 we could present a diagram, which is equivalent to Table 1. So weget the next result.

Theorem 5.4. The number and multiplicities of the finite singular points (de-scribed by the divisor DS(P,Q)) for non-degenerate quadratic systems (4.1) is givenby the diagram presented in Figure 1.

We are interested in a global characterization of the singularities (finite and in-finite) of the family of quadratic systems. More precisely we would like to extendthe diagram of Figure 1 adding the infinite singularities (their number and multi-plicities) and then including the types of all these singularities. Moreover we wishto distinguish the weak singularities (if it is the case) as well as their order. This isone of the motivations why we consider again the class of quadratic systems with


Figure 1. Diagram for Finite Singularities of Quadratic systems

centers as well as the class of Hamiltonian systems, the topological classificationsof which could be found in articles [15, 13, 14] and [16], respectively.

On the other hand we would like to reveal the main affine invariant polynomialsassociated to the singularities of quadratic systems, having a transparent geomet-rical meaning. And it is clear that all the conditions we need have to be based onthe invariant polynomials contained in the diagram of Figure 1.


Thus in this article new geometrical more transparent affine invariant condi-tions for distinguishing topological phase portraits of the two mentioned familiesof quadratic systems are simultaneously constructed. For this purpose we need thefollowing GL-invariant polynomials constructed in tensorial form in [29] (we keepthe respective notations)

I1 = aαα, I2 = aα

βaβα, I3 = aα

p aβαqa

γβγεpq, I4 = aα

p aββqa

γαγεpq,

I5 = aαp aβ

γqaγαβεpq, I6 = aα

p aβγaγ

αqaδβδε

pq, I7 = aαpra

βαqa

γβsa

δγδε

pqεrs,

I8 = aαpra

βαqa

γδsa

δβγεpqεrs, I9 = aα

praββqa

γγsa

δβγεpqεrs,

I10 = aαp aβ

δ aγνaδ

αqaνβγεpq, I13 = aα

p aβqra

γγsa

δαβaν

δνεpqεrs,

I16 = aαp aβ

r aγδ aδ

αqaνβsa

τγµaµ

ντεpqεrs, I18 = aαaqapαεpq,

I19 = aαaβγaγ

αβ , I21 = aαaβaqapαβεpq, I23 = aαaβaγ

αδaδβγ ,

I24 = aαaβδ aγ

αaδβγ , I28 = aαaβaγ

δ aδγµaµ

αβ , I30 = aαaβpaγ

βqaδγµaµ

αδεpq,

(5.3)

I33 = aαaβaγaδαβaµ

γνaνδµ, I35 = aαaβ

paγαaδ

βqaµγνaν

δµεpq,

K1 = aααβxβ , K2 = ap

αxαxqεpq, K3 = aαβaβ

αγxγ , K5 = apαβxαxβxqεpq,

K7 = aαβγaβ

αδxγxδ K11 = ap

αaαβγxβxγxqεpq, K12 = aα

βaβαγaγ

δµxδxµ,

K14 = aαp aβ

αqaγβδa

δγµxµεpq, K21 = apxqεpq, K22 = aαap

αxqεpq,

K23 = apaqαβxαxβεpq, K27 = aαaβ

αγaγβδx

δ, K31 = aαaβαγaγ

βδaδµνxµxν ,

(5.4)

where ε11 = ε22 = ε11 = ε22 = 0, ε12 = −ε21 = ε12 = −ε21 = 1. We note thatthe expressions for the above invariants are associated to the tensor notation forquadratic systems (4.1) (see [29])

dxj

dt= aj + aj

αxα + ajαβxαxβ , (j, α, β = 1, 2);

a1 = a00, a11 = a10, a1

2 = a01, a111 = a20, a1

22 = a02,

a2 = b00, a21 = b10, a2

2 = b01, a211 = b20, a2

22 = b02,

a112 = a1

21 = a11, a212 = a2

21 = b11.

6. The family of quadratic systems with centers

The proof of Theorem 1.1 is based on the classification of the family of quadraticsystems with centers given in [30]. Using the expressions (5.3) of [30] we get thefollowing GL-invariants (we keep the respective notations adding only the ”hat”)

α = I22I8 − 28I2I

25 + 6I5I10, β = 4I2

4 − 3I2I9 − 4I3I4,

γ =3I24

(2I3I4 + I2I9), δ = 27I8 − I9 − 18I7,

ξ = I2I5(I2I5 + 2I10)− 4I210 − I3

2I8.

(6.1)

We note that in [30] the expressions of the invariants ξ and δ are used directly, butwe set these notations for compactness.

According to [30] (see Lemmas 2-5) the next result follows.


Table 2.

Conditions for the existence of a center [statement (b) of Proposition 4.1]:T3 6= 0, T4 = F1 = F2 = F3F4 = 0, T3F < 0, (b4)

Additional conditions for configurations Configurationof singularities No.

µ0 6= 0

D < 0eK < 0 c, s, s, s ; N, N, N 1

eK > 0 c, s, n, n ; S, S, N 2

D > 0η < 0 c, f ; S 3

η > 0 c, n ; S, S, N 4

D = 0 c, n, sn(2) ; S,`02

´SN 5

µ0 = 0 c, s ; N,`11

´SN,

`11

´SN 6

Conditions for the existence of a center [statement (c) of Proposition 4.1]:T4 = T3 = 0, T2 6= 0, (c5) ∪ (c6)

µ0 < 0

D < 0

η < 0 c, $, n, n ; S 7

η > 0eK < 0 c, $, s, s ; N, N, N 8

eK > 0 c, $, n, n ; S, S, N 9

η = 0 c, $, n, n ;`03

´S 10

D > 0

η < 0 c, c ; S 11

η > 0 c, c ; S, S, N 12

η = 0 c, c ;`03

´S 13

µ0 > 0

D < 0 c, c, s, s ; N 14

D > 0

η < 0 c, $ ; N 15

η > 0 c, $ ; S, N, N 16

η = 0 c, $ ;`03

Ń 17

µ0 = 0

η < 0eK 6= 0

µ2 < 0 c, c ;`21

´S 18

µ2 > 0 c, $ ;`21

Ń 19

eK = 0 c, $ ; N 20

η > 0eK 6= 0 c, $ ;

`21

´S, N, N 21

eK = 0 c, $ ; N,`11

´SN,

`11

´SN 22

Proposition 6.1. Assume that for a quadratic system with the singular point (0, 0)the conditions I1 = I6 = 0 and I2 < 0 hold. Then this system has a center at (0, 0)and via a linear transformation could be brought to one of the canonical forms belowif and only if the respective additional GL-invariant conditions hold

(S(c)1 )

{x = −y + gx2 − xy, (g 6= 0),y = x + x2 + 3gxy − 2y2,

⇐⇒ I3I13 6= 0, 5I3 − 2I4 = 13I3 − 10I5 = 0;

(S(c)2 )

{x = y + 2nxy, (wm 6= 0),y = −x + lx2 + 2mxy − ly2,

⇐⇒ I3 = 0, I13 6= 0;


Table 2 (continued)

Conditions for the existence of a center [statement (e) of Proposition 4.1]:T4 = T3 = T2 = 0, σ 6= 0, (e4)

Additional conditions for configurations Configurationof singularities No.

µ0 < 0

η < 0 c, bes(3) ; S 23

η > 0 c, bes(3) ; S, S, N 24

η = 0 c, bes(3) ;`03

´S 25

µ0 > 0 c, s(3) ; N 26

µ0 = 0

µ1 6= 0

D < 0 c, s, s ; N,c12

´P E P −H 27

D > 0

eN ≤ 0 c ; S,c12

´P E P −H 28

eN > 0

L < 0 c ; S,c12

´P E P − P H P 29

L > 0 c ; N,c12

´H −H H H 30

L = 0 c ; [∞, S] 31

µ1 = 0µ3 6= 0

L 6= 0 c ; N,`32

´H −H H H 32

L = 0 c ;`21

´S,

c12

´P E P −H 33

µ3 = 0 c,`

[|]; ∅´;

` [|]; ∅

´34

Conditions for the existence of a center [statement (f) of Proposition 4.1]:Hamiltonian systems ⇒ σ = 0, (f1)–(f3), (f5), (f7)

µ0 < 0 c, $, $, $ ; N, N, N 35

µ0 > 0

D < 0 c, c, $, $ ; N 36

D > 0 c, $ ; N 37

D = 0T 6= 0 c, $, bcp(2) ; N 38

T = 0 c, bs(3) ; N 39

µ0 = 0µ1 6= 0 c, $, $ ; N,

c12

´PEP −H 40

µ1 = 0 c, $ ;c23

Ń 41

(S(c)3 )

{x = y + 2(1− e)xy,

y = −x + dx2 + ey2,⇐⇒ I13 = 0, I4 6= 0;

(S(c)4 )

{x = y + 2cxy + by2,

y = −x− ax2 − cy2, c ∈ {0, 1/2}⇐⇒ I4 = 0;

where w = m2(2n− l)− (n− l)2(2n + l).

Proof of Theorem 1.1. We shall consider each one of the systems (S(c)1 )− (S(c)

4 )and will compare the GL-invariant conditions [30] with the affine invariant onesgiven by Tables 2 and 3.

6.1. The family of systems (S(c)1 ). For these systems we calculate the respective

GL-invariants

I13 = 125g(1 + g2)/8, I3 = 5(1 + g2)/2 (6.2)


Table 3.

Con-figu-

ration

Phaseportrait

Con-figu-

ration

Phaseportrait

Con-figu-

ration

Phaseportrait

1 Vul 10 15 Vul 2 30 Vul 13

2 Vul 27 16 Vul 19 31 Vul 15

3 Vul 30 17 Vul 2 32 Vul 13

4 Vul 32 18 Vul 20 33 Vul 12

5 Vul 31 19 Vul 2 34 Vul 29

6 Vul 17 20 Vul 2

35

Vul 11 if B1 6= 0

7 Vul 25 21 Vul 19 Vul 9 if B1 = 0, B3B4 < 0

Vul 9 if B3B5 < 0 Vul 18 if B3B5 < 0 Vul 8 if B1 = 0, B3B4 > 0

8 Vul 8 if B3B5 > 0 22 Vul 16 if B3B5 > 0 Vul 10 if B1 = B3 = 0

Vul 10 if B3 = 0 Vul 17 if B3 = 036

Vul 4 if B1 6= 0

Vul 28 if B3B5 < 023

Vul 22 if θ1 < 0 Vul 3 if B1 = 0

9 Vul 26 if B3B5 > 0 Vul 23 if θ1 ≥ 0 37 Vul 2

Vul 27 if B3 = 0 24 Vul 24 38 Vul 7

10 Vul 25 25 Vul 23 39 Vul 2

11 Vul 20 26 Vul 240

Vul 6 if B1 6= 0

12 Vul 21 27 Vul 5 Vul 5 if B1 = 0

13 Vul 20 28 Vul 12 41 Vul 214 Vul 3 29 Vul 14

and the affine invariant polynomials

T4 = F1 = F2 = F4 = 0, T3 = −125g(1 + g2), F3 = 625g2(1 + g2)2,

µ0 = −2(1 + g2), D = 5184g2(1 + g2), η = 4(1 + g2).(6.3)

According to [30] in the case I13 6= 0 the phase portrait of systems (S(c)1 ) is given

by Vul 32 and hence we have the configuration of the singularities c, n; S, S, N .On the other hand the condition I13 6= 0 implies T3 6= 0 and as F1 = F2 =

F3F4 = 0, the conditions provided by the statement (b4) of Proposition 4.1 aresatisfied. Moreover, as µ0 6= 0, D > 0 and η > 0, we obtain the respectiveconditions given by Table 2 (row No. 4).

6.2. The family of systems (S(c)2 ). In this case calculations yield:

I13 = m[m2(2n− l)− (n− l)2(2n + l)

], I9 − I8 = 4ln(l2 + m2 + n2),

δ = −8(l + 2n)2(l2 + m2 + 2ln).(6.4)

According to [30] (see Lemma 4) the phase portrait (and this yields the respectiveconfiguration of singularities) of a system from the family (S(c)

2 ) is determined bythe following GL-invariant conditions, respectively

I9 − I8 > 0 ⇔ Vul10 ⇒ c, s, s, s;N,N, N ;I9 − I8 = 0 ⇔ Vul17 ⇒ c, s ; N,

(11

)SN,

(11

)SN ;

I9 − I8 < 0, δ < 0 ⇔ Vul27 ⇒ c, s, n, n;S, S,N ;I9 − I8 < 0, δ > 0 ⇔ Vul30 ⇒ c, f ;S;I9 − I8 < 0, δ = 0 ⇔ Vul31 ⇒ c, n, sn(2);N,

(02

)SN.

(6.5)


On the other hand calculations yield

T4 = F1 = F2 = F3 = 0, T3F = −8m2[m2(2n− l)− (n− l)2(2n + l)

]2,

µ0 = −4l2n2, D = −192(l + 2n)2(l2 + m2 + 2ln) = −48η, K = −4ln(x2 + y2).(6.6)

We observe that the condition I13 6= 0 implies T3F < 0 and as F1 = F2 = F3F4 = 0we conclude that the conditions provided by statement (b4) of Proposition 4.1 arefulfilled. Moreover comparing (6.4) and (6.6) if Dµ0 6= 0 we obtain

sign(I9 − I8) = − sign(K), sign(δ) = sign(D) = − sign(η),

and I9 − I8 = 0 (respectively δ = 0) if and only if µ0 = 0 (respectively D = 0). Sotaking into consideration that the condition K < 0 implies D < 0, we obviouslyarrive to the conditions provided by Table 2 (the case of statement (b) of Proposition4.1).

6.3. The family of systems (S(c)3 ). For these systems calculations yield

T4 = T3 = T1 = F = F1 = 0, T2 = 4d(d + 2− 2e),

B = −2, H = 4d(1− e), σ = 2y.(6.7)

Therefore according to Proposition 4.1 for a system (S(c)3 ) could be satisfied either

the conditions of statement (c) (if T2 6= 0) or of statement (e) (if T2 = 0). We shallconsider each one of these possibilities.

6.3.1. The case T2 6= 0. Following [30, Lemma 3] for systems (S(c)3 ) we calculate

I4 = −1, I3 = −(d + e), I9 = d, β = 2(d + 2− 2e), γ = 6e, (6.8)

and therefore the condition T2 6= 0 is equivalent to I9β 6= 0. It was above mentionedthat the conditions of statement (c) (Proposition 4.1) are satisfied in this case; i.e.,there should be two weak singularities on the phase plane of these systems. Soaccording to [30] (see the proof of Lemma 3) the phase portrait (and this yields therespective configuration of singularities) of a system from the family (S(c)

3 ) with thecondition I9β 6= 0 is determined by the following GL-invariant conditions

γI9 > 0, βγ < 0 ⇔ Vul3 ⇒ c, c, s, s ;N ;

{γI9 > 0, βγ > 0, I9(4− γ) ≤ 0,

or γ = 0, β < 0⇔ Vul2 ⇒

c, $ ;N if γ(γ − 4)(γ − 6) 6= 0;

c, $ ;(21

)N if γ = 0;

c, $ ;(03

)N if γ = 4;

c, $ ;N if γ = 6;

I9 < 0, 0 ≤ γ ≤ 4, β > 0 ⇔ Vul20 ⇒

c, c ;S if γ(γ − 4) 6= 0;

c, c ;(21

)S if γ = 0;

c, c ;(03

)S if γ = 4;

I9 < 0, γ > 4, β > 0 ⇔ Vul21 ⇒ c, c ;S, S,N ; (6.9)

I9 > 0, 0 ≤ γ < 4 ⇔ Vul19 ⇒

{c, $ ;S, N,N if γ 6= 0;

c, $ ;(21

)S, N,N if γ = 0;


β < 0, 0 < γ ≤ 4 ⇔ Vul25 ⇒

{c, $, n, n ;S if γ 6= 4;

c, $, n, n ;(03

)S if γ = 4;

and(i) γI9 < 0, γ(γ − 6) > 0 ⇔ Vul8,Vul9,Vul10 ⇒ c, $, s, s ;N,N, N ;

(ii) I9 < 0, γ = 6 ⇔ Vul16,Vul17,Vul18 ⇒ c, $ ; N,

(11

)SN,

(11

)SN ;

(iii) β < 0, 4 < γ < 6 ⇔ Vul26,Vul27,Vul28 ⇒ c, $, n, n ;S, S,N.

(6.10)

On the other hand for systems (S(c)3 ) we calculate

µ0 = 4de(e− 1)2, D = 192e(d + 2− 2e)3, η = 4d(2− 3e)3,

K = 4(e− 1)(dx2 − ey2), µ1 = 4e(e− 1)(d + e− 1)y,

µ2

∣∣e=0

= d(d + 2)x2, µ2

∣∣e=1

= d(dx2 + y2).

(6.11)

So considering (6.8) and (6.11) if Dµ0η 6= 0 we obtain

sign(µ0) = sign(γI9), sign(D) = sign(βγ), sign(η) = sign(I9(4− γ)),

and due to T2 6= 0 we have that µ0 = 0 (respectively D = 0; η = 0) if and only ifγ(γ − 6) = 0 (respectively γ = 0; γ = 4). Therefore it is not too hard to determinethat in cases (6.9) (when we have the unique phase portrait) the conditions fromTable 2 (the case of statement (c) of Proposition 4.1) are equivalent to the respectiveconditions from (6.9).

We consider now the remaining cases (6.10). According to [30] the phase por-traits Vul 8, Vul9, Vul10 (respectively Vul 16,Vul17,Vul18; Vul 26,Vul27,Vul28) aredistinguished via the GL-invariant γI3I4. More precisely in the mentioned casesthe phase portrait corresponds to Vul 8 (respectively Vul 16; Vul 26) if γI3I4 > 0;Vul 9 (respectively Vul 18; Vul 28) if γI3I4 < 0 and it corresponds to Vul 10 (respec-tively Vul 17; Vul 27) if I3 = 0.

On the other hand for systems (S(c)3 ) we calculate

B3 = −6(2 + d− 3e)(d + e)x3y, B3B5 = 288d(2 + d− 3e)e(d + e)x4y2.

We claim that in all three cases (6.10), if B3 6= 0 then we have

sign(B3B5) = sign(γI3I4) = sign(e(d + e)

), (6.12)

and B3 = 0 if and only if I3 = 0 (i.e. d + e = 0). To prove this claim we shallconsider each one of the cases (i)− (iii) from (6.10).

Case (i). Considering (6.8) we have de < 0, e(e− 1) > 0 and herein it can easilybe detected that sign(d + 2− 3e) = − sign(e) and this leads to (6.12).

Case (ii). As e = 1 we have B3B5 = 288d(d − 1)(1 + d)x4y2 and due to d < 0this evidently implies (6.12).

Case (iii). In this case considering (6.8) we have 2/3 < e < 1 and d+2−2e < 0.Therefore d < 0 and 2− 3e < 0 that gives d+2− 3e < 0. As e > 0 we again obtain(6.12).

It remains to note that due to the conditions discussed above in all three caseswe can have B3 = 0 if and only if d + e = 0 (i.e. I3 = 0).

Thus our claim is proved and obviously we arrive to the conditions of Table 3corresponding to the configurations 8, 9 and 22 respectively.


6.3.2. The case T2 = 0Then d(d+2−2e) = 0 and considering (6.8) for systems (S(c)

3 ) we have I9β = 0.We recall that by Proposition 4.1 in this case for a system (S(c)

3 ) has to be satisfiedthe conditions of the statement (e), i.e. besides the center we could not haveanother weak singularity. So according to [30] (see the proof of Lemma 3) thephase portrait (and hence, the respective configuration of singularities) of a systemfrom the family (S(c)

3 ) with the condition I9β = 0 is determined by the followingGL-invariant conditions

I9 = 0, γ(γ − 6) > 0 ⇔ Vul5 ⇒ c, s, s ;N,

(12

)P E P −H;

I9 = 0, 0 ≤ γ ≤ 3 ⇔ Vul12 ⇒

{c ;S,

(12

)P E P −H if γ 6= 0;

c ;(21

)S,

(12

)P E P −H if γ = 0;

I9 = 0, 3 < γ < 4 ⇔ Vul14 ⇒ c ;S,

(12

)P E P − P H P ;

I9 = 0, γ = 4 ⇔ Vul15 ⇒ c ; [∞, S];

I9 = 0, 4 < γ ≤ 6 ⇔ Vul13 ⇒

{c ;N,

(12

)H −HHH if γ 6= 6;

c ;N,(32

)H −HHH if γ = 6;

(6.13)

andβ = 0, γI9 > 0, I9(4− γ) < 0 ⇔ Vul2 ⇒ c, s(3) ;N ;

β = 0, 0 < γ < 3 ⇔ Vul22 ⇒ c, es(3) ;S;

β = 0, 3 ≤ γ ≤ 4 ⇔ Vul23 ⇒

{c, es(3) ;S if γ 6= 4;

c, es(3) ;(03

)S if γ = 4;

β = 0, 4 < γ < 6 ⇔ Vul24 ⇒ c, es(3) ;S, S,N ;

β = 0, γ = 0 ⇔ Vul29 ⇒ c,(

[|]; ∅);

( [|]; ∅

).

(6.14)

We remark that by (6.8) the condition I9 = 0 gives d = 0 whereas the conditionβ = 0 gives d = 2(e− 1).

(1) Assume first I9 = 0, i.e. d = 0. Then for systems (S(c)3 ) we obtain

µ0 = η = 0, D = −1536e(e− 1)3, N = 4(1− e)(2e− 1)y2,

L = 8e(3e− 2)y2, µ1 = 4e(e− 1)2y, µ3 = 2(1− e)x2y + ey3.(6.15)

As γ = 6e (see (6.8)) this implies γ − 6 = 6(e− 1), γ − 4 = 2(3e− 2) and γ − 3 =3(2e− 1). So if γ(γ − 3)(γ − 4)(γ − 6) 6= 0 then we have

sign(D) = − sign(γ(γ − 6)

), sign(N)

∣∣{D>0} = sign(γ − 3

),

sign(L)∣∣{D>0, eN>0} = sign(γ − 4

).

(6.16)

Moreover if µ1 6= 0 then N = 0 (respectively L = 0) if and only if γ = 3 (respectivelyγ = 4). Therefore to determine the cases (6.13) (when I9 = 0) the conditions ofTable 2 (the case of statement (e) of Proposition 4.1) are equivalent to the respectiveconditions of (6.13).


(2) Suppose now β = 0 and I9 6= 0. Then d = 2(e − 1) 6= 0 and for systems(S(c)

3 ) we have

µ0 = 8e(e− 1)3, η = 8(e− 1)(2− 3e)3, µ1 = 12e(e− 1)2y,

µ3 = ey3, θ1 = 128(e− 1)(2e− 1)(3e− 2)(5− 6e).(6.17)

We observe that if µ0 6= 0 then

sign(µ0) = sign(γ(γ − 6)

), sign(η)

∣∣{µ0<0} = sign(γ − 4

),

sign(θ1)∣∣{µ0<0,η<0} = sign(γ − 3

).

(6.18)

We claim that the conditions γI9 > 0 and I9(4 − γ) < 0 corresponding to thephase portrait Vul2 (see (6.14)) are equivalent to µ0 > 0. Indeed, as I9 = d =2(e − 1) 6= 0 we obtain γI9 = 12e(e − 1) and hence the condition γI9 > 0 isequivalent to µ0 > 0. It remains to note that in the case e(e − 1) > 0 we haveI9(4− γ) = −4(e− 1)(3e− 2) < 0. So 3e− 2 6= 0 (i.e. γ 6= 4) and we arrive to therespective conditions from Table 2.

Considering the remaining cases (6.14) corresponding to the condition β = 0and (6.18) we arrive to the respective conditions provided by Table 2 (the case ofstatement (e) of Proposition 4.1).

6.4. The family of systems (S(c)4 ). For these systems we have σ = 0; i.e., this is

a class of Hamiltonian systems with center. Moreover any elemental point which isnot a center must be an integrable saddle. Calculations yield

α = 8(a− 2c)2(b2 − 4ac + 8c2), I8 = 2a(ab2 + 4c3),

I10 = −b2 − (a + c)2, I16 = b(3ac2 − a3 + ab2 + 2c3).(6.19)

According to [30] (see the proof of Lemma 2) the phase portrait (and hence, therespective configuration of singularities) of a quadratic system from the family (S(c)

4 )is determined by the following GL-invariant conditions{

α < 0, orα = I16 = 0

⇔ Vul2 ⇒

c, $ ;N if α 6= 0;c, s(3);N if α = 0, I8 6= 0;

c, $ ;(23

)N if α = 0, I8 = 0;

α > 0, I8 < 0 ⇔ Vul8,Vul9,Vul10,Vul11 ⇒ c, $, $, $ ;N,N, N ;α > 0, I8 > 0 ⇔ Vul3,Vul4 ⇒ c, c, $, $ ;N ;

α > 0, I8 = 0 ⇔ Vul5,Vul6 ⇒ c, $, $ ;N,

(12

)PEP −H;

α = 0, I16 6= 0 ⇔ Vul7 ⇒ c, $, cp(2) ;N.

(6.20)

On the other hand for systems (S(c)4 ) calculations yield

µ0 = a(ab2 + 4c3) = I8/2, D = −48(a− 2c)2(b2 − 4ac + 8c2) = −6α,

µ1 = 2ab(a− c)x− 2(ab2 + 2ac2 + 2c3)y.(6.21)

We observe that the condition I8 < 0 implies c 6= 0 (then by Proposition 6.1 wehave c = 1/2) and a < 0. In this case we evidently obtain α > 0. Similarly thecondition α < 0 gives c 6= 0 (i.e. c = 1/2) and a > 1 + b2/2 and this impliesI8 > 0. Herein we conclude that in the case I8α 6= 0 (i.e. µ0D 6= 0) the conditions


provided by Table 2 (the case of statement (f) of Proposition 4.1) for distinguishingthe configurations of the singularities are equivalent to the respective conditions of(6.20).

Assume now I8α = 0 (i.e. µ0D = 0).(1) If I8 = 0 then by (6.19) we have a(ab2 +4c3) = 0 and then µ0 = 0. We claim

that in the case α 6= 0 we obtain α > 0 and this is equivalent to µ1 6= 0. Indeed asthe condition α 6= 0 implies (a2 + c2)(b2 + c2) 6= 0, we conclude that the conditionI8 = 0 gives a ≤ 0 and c 6= 0. So c = 1/2 and we obtain α > 0. On the other handsince a − c 6= 0 we obtain that µ1 = 0 if and only if ab = a + c = 0 but in thiscase the condition I8 = 0 implies α = 0. So our claim is proved and this shows theequivalence of the respective conditions of Table 2 and (6.20).

Assume I8 = 0 = α. Then considering (6.20) we obtain portrait Vul2 and theconfiguration of singularities indicated in the row 41 of Table 2. It remains toobserve that the condition above is equivalent to µ0 = µ1 = 0.

(2) Suppose now α = 0 and I8 6= 0. This implies µ0 6= 0 and D = 0, i.e.(a− 2c)(b2 − 4ac + 8c2)=0. So we obtain

T = −48b2c4y2(2cx + by)2(bx− cy)2, I16 = 2b3c, µ0 = 4c2(b2 + 2c2),

if a = 2c and

T = − 3b2

4096c6(b2 + 4c2)4y2(bx− 2cy)2(b2x + 8c2x + 2bcy)2,

I16 = − 164c3

b3(b2 + 4c2)2, µ0 =1

16c2(b2 + 4c2)2(b2 + 8c2),

if a = (b2 + 8c2)/(4c) (we note that c 6= 0 due to I8 6= 0).Thus clearly if α = 0 and I8 6= 0 then the condition I16 = 0 is equivalent to

T = 0 and as µ0 > 0 we obtain that the respective conditions provided by Table 2(see rows 38 and 39) are equivalent to the corresponding conditions from (6.20).

To finish the proof of Theorem 1 it remains to examine the conditions for distin-guishing the different phase portraits which correspond to the same configurationof singularities. We have three groups of such phase portraits: (i) Vul8 − Vul11;(ii) Vul3,Vul4 and (iii) Vul5,Vul6. According to [31] quadratic systems (S(c)

4 ) pos-sess one of the mentioned phase portraits if and only if the following conditions arefulfilled

α > 0, I8 > 0 ⇒

{Vul3 if I16 = 0;Vul4 if I16 6= 0;

α > 0, I8 = 0 ⇒

{Vul5 if I16 = 0;Vul6 if I16 6= 0;

α > 0, I8 < 0 ⇒

Vul8 if I10 6= 0, I16 = 0, ξ > 0;Vul9 if I10 6= 0, I16 = 0, ξ < 0;Vul10 if I10 = 0;Vul11 if I16 6= 0.

(6.22)

On the other hand for systems (S(c)4 ) we have

B1 = −I16[b2 + (a− 3c)2], I16 = b(3ac2 − a3 + ab2 + 2c3),

B3 = −3abx4 − 6(a− 3c)(a + c)x3y + 18bcx2y2 + 6b2xy3 + 3b(a− 2c)y4,


and hence the condition I16 = 0 is equivalent to B1 = 0. Herein we arrive to theconditions provided by Table 3 for the portraits Vul3 −Vul6 respectively.

Next we examine the conditions for the phase portraits Vul8 − Vul11. First weobserve that by (6.19) the condition I10 = 0 yields b = a + c = 0 and this impliesI16 = B3 = B1 = 0. Therefore in the case I16 6= 0 (this is equivalent with B1 6= 0)we get Vul 11 and in the case I10 = 0 (then B3 = 0) we obtain Vul 10 (see Table 3,configuration number 35). So it remains to consider the phase portraits Vul 8 andVul 9. We claim that in the case I16 = 0 we have sign(ξ) = sign(B3B4). Indeedassuming I16 = 0 we shall examine two cases, b = 0 and b 6= 0.

(1) If b = 0 (then I16 = 0) a straightforward calculation for systems (S(c)4 ) yields

ξ = −4(a− c)(a + c)3, B3B4 = −192(a− 3c)3c4(a + c)3x4y2, I8 = 8ac3,

and as I8 < 0 (i.e. ac < 0 ) we get (a − c)(a − 3c) > 0. So clearly our claim isproved in this case.

(2) Assume now b 6= 0. Then I8 < 0 gives ac < 0 and hence we can set anew parameter u as follows u2 = (a − 2c)/a. Herein we have c = a(1 − u2)/2 andcalculation gives

I16 = ab(2b− 3au + au3)(2b + 3au− au3)/4.

Hence due to ab 6= 0 the condition I16 = 0 gives b = ±au(u2 − 3)/2 and then wecalculate

ξ = −a4(u2 − 1)(u2 − 3)3(1 + u2)2/2, I8 = a4(2− u2)(1 + u2)2/4,

B3B4 = −3a10(u2 − 3)3(1 + u2)5(ux± y)2(uy ∓ x)4/16.

As I8 < 0 we have u2 − 2 > 0 and this implies u2 − 1 > 0. Therefore sign(ξ) =sign(B3B4), i.e. our claim is valid and we arrive to the respective conditions givenby Table 3 in the considered case.

As all the cases are examined Theorem 1.1 is proved. �

7. The family of Hamiltonian quadratic systems

In this section to prove Theorem 1.2 we need the following invariant polyno-mials defined in [16] using the invariant polynomials (5.3) and (5.4) (we keep the


respective notations adding only the “hat”)

2µ = I8, H = −K14, 4G = −5I2K7 + 2I5K2 + 4K23 + 8K31,

2F = −I2K11 − 4I19K5 + 4K2K27, V = K11K22 + K223,

R = 3H2 − 2Gµ, S = 2F Hµ2 + G2µ2 − 4GH2µ + 3H4 − 4µ3V ,

P = G2 − 6F H + 12µV , U = F 2 − 4GV ,

T = 9F 2µ2 − 14F GHµ + 12F H3 + 2G3µ− 2G2H2 − 8Gµ2V + 12H2µV ,

W1 = K22 − 4K5K21, W2 = −K2K12 − 2K3K11 + 4K5K27 + 6K7K23,

E1 = 4I10 − 5I2I5 − 24I30, E2 = I22 + 32I23 − 16I24,

E3 = I32 + 32(12I5I18 − 3I2

19 + 16I33),

32D = 4I25 (3I2E2 − E3)− E2(4I5E1 + I8E2),

Tc = (9I35 + I8E1)2 + 9I2

5 (I28 E2 − 3I4

5 ) + I38 (3I2E2 − E3),

Uc = 3I8I19(3I2I5 + 16I30)− 2I8(3I2I35 + 24I5I28 + 16I8I21)

+3I25 (I5I19 + 10I35)− 2I16(2I2I5 − I10 + 12I30).

(7.1)

The proof of Theorem 1.2 is based on the classifications of quadratic Hamiltoniansystems given in [1] and [16]. We use the notations of [1] whether the system has acenter as Vul# or if not as Ham#. In the first paper the global phase portraits ofthis family were studied. In the second one there are determined the affine invariantcriteria for the realization of each one of the 28 possible topologically distinct phaseportraits constructed in [1]. That is, the phase portraits of the systems

dx

dt=

∂H

∂y,

dy

dt= −∂H

∂x, (7.2)

where H(x, y) is a polynomial of degree 3 in the variables x and y over R.According to the paper [4] for the quadratic Hamiltonian systems we have

H(x, y) =2∑

j=0

1j + 1

Cj(x, y), (7.3)

where Cj(x, y), j = 0, 1, 2 are the polynomials (4.3). So the 3rd degree homogeneouspart of the polynomial H(x, y) is the polynomial H3(x, y) = C2(x, y)/3. As it isshown in [1] via linear transformations the non-zero form H3(x, y) can be derivedto one of the 4 canonical forms

a) x(x2 − y2); b) (x3 + y3)/3; c) x2y; d) x3/3.

By (4.5) we observe that the invariant polynomials η and M are respectively thediscriminant and the Hessian of the binary form C2(x, y). So considering also [16]we arrive to the next proposition.

Proposition 7.1. Assume that for a quadratic system the condition ∂∂xP (x, y) +

∂∂y Q(x, y) ≡ 0 holds, i.e. it is Hamiltonian. Then this system could be brought viaan affine transformation and time rescaling to one of the canonical forms below if


Table 4.

Affine invariant conditions for configurations Configurationof singularities No.

µ0 < 0

D < 0 $, $, $, c ; N, N, N 1

D > 0 $, $ ; N, N, N 2

D = 0

T 6= 0 $, $, bcp(2) ; N, N, N 3

T = 0R 6= 0 $, bs(3) ; N, N, N 4

R = 0 (hhhhhh)(4) ; N, N, N 5

µ0 > 0

D < 0(R > 0)&(S > 0) $, $, c, c ; N 6

(R ≤ 0) ∨ (S ≤ 0) ∅; N 7

D > 0 $, c ; N 8

D = 0

T < 0 $, c, bcp(2) ; N 9

T > 0 bcp(2) ; N 10

T = 0

PR < 0 ∅; N 11

PR > 0 bcp(2), bcp(2) ; N 12

PR = 0R 6= 0 c, bs(3) ; N 13

R = 0 (hh)(4) ; N 14

µ0 = 0

µ1 6= 0

D < 0 $, $, c ; N,c12

´PEP −H 15

D > 0 $ ; N,c12

´PEP −H 16

D = 0 P 6= 0 $, bcp(2) ; N,c12

´P E P −H 17

P = 0 bs(3) ; N,c12

´PEP −H 18

µ1 = 0

µ2 6= 0

U < 0fM 6= 0 ∅; N,

`22

´H −H 19

fM = 0 ∅;c23

Ń 20

U > 0fM 6= 0 $, $ ; N,

`22

´P E P − P E P 21

fM = 0 $, c ;c23

Ń 22

U = 0 bcp(2) ;c23

Ń 23

µ2 = 0

µ3 6= 0 $ ;`33

´PEPEP − P 24

µ3 = 0,µ4 6= 0

fM 6= 0 ∅; N,`42

´PHP − PHP 25

fM = 0 ∅;`43

Ń 26

µ3 = µ4 = 0, fM 6= 0`

[|]; s´; N,

` [|]; N

´27

µ3 = 0,µ4 = 0,fM = 0

B6 < 0`

[‖c]; ∅´;

` [‖c]; N

´28

B6 > 0`

[‖]; ∅´;

` [‖]; N

´29

B6 = 0`

[|2]; ∅´;

` [|2]; N

´30

and only if the respective conditions hold

(S(h)1 )

{x = α + bx + cy − 2xy,

y = β − ax− by − 3x2 + y2,⇔ η > 0;

(S(h)2 )

{x = α + bx + cy + y2,

y = β − ax− by − x2,⇔ η < 0;


Table 5.

Con-figu-

ration

Phaseportrait

Con-figu-

ration

Phaseportrait

Con-figu-

ration

Phaseportrait

1

Vul 11 if B1 6= 0 8 Vul 2 20 Ham11Vul 9 if B1 = 0, B3B4 < 0 9 Vul 7

21Ham20 if B3 6= 0

Vul 8 if B1 = 0, B3B4 > 0 10 Ham17 Ham21 if B3 = 0Vul 10 if B1 = B3 = 0 11 Ham11 22 Vul 2

2Ham26 if B1 = 0, B3B4 < 0 12 Ham28 23 Ham17

Ham25 ifn

B1 6= 0, orB1 = 0, B3B4 > 0 13 Vul 2 24 Ham18

3Ham27 if B1 6= 0 14 Ham17 25 Ham13Ham24 if B1 = 0

15Vul 6 if B1 6= 0 26 Ham11

4 Ham25 Vul 5 if B1 = 0 27 Ham14

5 Ham23 16 Ham19 28 Ham11

6Vul 4 if B1 6= 0 17 Ham22 29 Ham15

Vul 3 if B1 = 0 18 Ham19 30 Ham167 Ham11 19 Ham12

(S(h)3 )

{x = α + bx + cy + x2,

y = β − ax− by − 2xy,⇔ η = 0, M 6= 0;

(S(h)4 )

{x = α + bx + cy,

y = β − ax− by − x2,⇔ η = 0, M = 0.

Remark 7.2. For a quadratic Hamiltonian system the relation η = −27µ0 holds.This could be easy established via the frontal evaluations of the invariant polyno-mials η and µ0 for systems (S(h)

1 )− (S(h)4 ).

Proof of Theorem 1.2. In what follows we shall consider each one of the systems(S(h)

1 )− (S(h)4 ) and will show that the conditions given in [16] for distinguishing the

phase portraits of the respective systems, are equivalent with the affine invariantprovided in Tables 4 and 5.

7.1. The family of systems (S(h)1 ). In this case according to [16, Theorem 3]

the phase portrait (and this yields the respective configuration of singularities) of asystem of the family (S(h)

1 ) is determined by the following affine invariant conditions

D < 0, Uc = 0, Tc > 0 ⇔ Vul8 ⇒ $, $, $, c ;N,N, N ;

D < 0, Uc = 0, Tc < 0 ⇔ Vul9 ⇒ $, $, $, c ;N,N, N ;

D < 0, Uc = 0, Tc = 0 ⇔ Vul10 ⇒ $, $, $, c ;N,N, N ;

D < 0, Uc 6= 0 ⇔ Vul11 ⇒ $, $, $, c ;N,N, N ;

D = 0, Uc = 0, Tc = 0 ⇔ Ham23 ⇒ (hhhhhh)(4) ;N,N, N ;

D = 0, Uc = 0, Tc < 0 ⇔ Ham24 ⇒ $, $, cp(2) ;N,N, N ;{D > 0, Uc 6= 0, orD ≥ 0, Uc = 0, Tc > 0

⇔ Ham25 ⇒

{$, $ ;N,N, N if D 6= 0;$, s(3) ;N,N, N if D = 0;

D > 0, Uc = 0, Tc < 0 ⇔ Ham26 ⇒ $, $ ;N,N, N ;

D = 0, Uc 6= 0 ⇔ Ham27 ⇒ $, $, cp(2) ;N,N, N.

(7.4)


According to [1] systems (S(h)1 ) have at least one real singular point. So we may

assume α = β = 0 due to a translation and then calculations yield

D = 4(b2 − ac)2[−8a3c− (36c2 + b2)a2 + 6c(4b2 − 9c2)a + 3(b4 + 33b2c2 − 9c4)],

Uc = 108bc(a− 2b− 3c)(a + 2b− 3c)(a− 2b + c)(a + 2b + c),

Tc = 3456[(a + 2b + c)3(a− 2b− 3c)3 + 24bc(a + 2b + c)2(a− 2b− 3c)2

− 384b2c2(a + 2b + c)(a− 2b− 3c)− 4096b3c3],

µ0 = −12 < 0, D = 48D, B1 = Uc/2, (7.5)

B3 = 18(a2 − 4b2 − 2ac− 3c2)x3y − 18bc(3x4 + 6x2y2 − y4),

B4 = 288[(a2 − 4b2 − 2ac− 3c2)x− 8bcy][24bcx + (a2 − 4b2 − 2ac− 3c2)y].

Remark 7.3. We claim that in the case Uc = 0 we have sign( T c) = sign(B3B4)and T c = 0 if and only if B3 = 0.

Indeed assume Uc = 0. We observe that the change of variables (x, y, t) 7→(x,−y,−t) keeps the systems (S(h)

1 ) with α = β = 0 except the sign of the parameterb, which will be changed. Hence without loss of generality we could consider onlythe equality bc(a+2b+ c)(a− 2b− 3c) = 0. If bc 6= 0 then by (7.5) evidently we getT c = −21933b3c3. On the other hand for a = −2b− c as well as for a = 2b + 3c, weobtain B3B4 = −21234b3c3(x − y)4(3x + y)2 and B3 = −18bc(x − y)3(3x + y). Soour claim is proved in the considered case.

Assume now bc = 0. Then calculations yield

T c = 3456(a− 3c)3(a + c)3, B3B4 = 3 T cx4y2/2, B3 = 18(a− 3c)(a + c)x3y

if b = 0 and

Tc = 3456(a− 2b)3(a + 2b)3, B3B4 = 3Tcx4y2/2, B3 = 18(a− 2b)(a + 2b)x3y

if c = 0. Now evidently the proof of the claim is completed.

7.1.1. The case D < 0. Then D < 0 and considering (7.5) and Remark 7.3 weconclude that the conditions for phase portraits Vul8 − Vul11 given by Tables 4and 5 are equivalent to the respective conditions in (7.4).

7.1.2. The case D > 0. According to (7.4) in this case we could have only the phaseportraits Ham25 and Ham26. So considering (7.5) and Remark 7.3 we again arriveto the equivalence of the respective conditions from (7.4) and from Tables 4 and 5.

7.1.3. The case D = 0. By (7.4) we have: (i) the phase portraits Ham24 (if Uc =0, Tc < 0) and Ham27(if Uc 6= 0) with the same configuration having three finitesingularities (one double); (ii) the phase portrait Ham25 (if Uc = 0, Tc > 0)possessing only two real finite singularities (one triple); (iii) the phase portraitHam23 (if Uc = Tc = 0) possessing only one finite singularity of multiplicity four.Considering the diagram in Figure 1 and (7.5) we conclude that the conditions fordetermining the mentioned phase portraits from Tables 4 and 5 are equivalent tothe respective conditions from (7.4).


7.2. The family of systems (S(h)2 ). In this case according to [16, Theorem 4]

the phase portrait (and this yields the respective configuration of singularities) ofa system from the family (S(h)

2 ) is determined by the following affine invariantconditions

{D > 0, orD = T = P = 0, R 6= 0

⇔ Vul2 ⇒

{$, c ;N if D 6= 0;c, s(3) ;N if D = 0;

D < 0, R > 0, S > 0, Uc = 0 ⇔ Vul3 ⇒ $, $, c, c ;N ;

D < 0, R > 0, S > 0, Uc 6= 0 ⇔ Vul4 ⇒ $, $, c, c ;N ;

D = 0, T < 0 ⇔ Vul7 ⇒ $, c, cp(2) ;N ;{D < 0, (R ≤ 0) ∨ (S ≤ 0), orD = T = 0, P R < 0

⇔ Ham11 ⇒ ∅;N ;{D = 0, T > 0, orD = T = P = R = 0

⇔ Ham17 ⇒

{cp(2) ;N if T 6= 0;(hh)(4) ; N if T = 0;

D = T = 0, P R > 0 ⇔ Ham28 ⇒ cp(2), cp(2) ;N.

(7.6)

Taking into account (7.1) and (5.2) a straightforward calculation gives for systems(S(h)

2 )

µ0 = 1, D = D/48, T = T/6, R = R/4, S = S/48, P = P, Uc = 2B1.

(7.7)Herein considering the diagram from Figure 1, it is easy to observe that the condi-tions for determining the phase portraits from Tables 4 and 5 corresponding to thecase µ0 > 0 are equivalent to the respective conditions from (7.6).

7.3. The family of systems (S(h)3 ). For this family of systems we have η = µ0 = 0

and M 6= 0. According to [28, Table 2] at infinity we have two real singularities oftotal multiplicity at least four. So in order to determine exactly the configurationof the singularities at infinity we shall use the classification of the behavior of thetrajectories in the neighbourhood at infinity of quadratic differential systems, givenin [28].

In this order of ideas we need the following additional invariant polynomials,defined in [28]

κ(a) = (M, K)(2), κ1(a) = (M, C1)(2),

K1(a, x, y) = p1(x, y)q2(x, y)− p2(x, y)q1(x, y),

K2(a, x, y) = 4(T2, ω)(1) + 3D1(C1, ω)(1) − ω(16T3 + 3T4/2 + 3D2

1

),

K3(a, x, y) = C22 (4T3 + 3T4) + C2(3C0K − 2C1T7) + K1(3K1 − C1D2),

(7.8)


where ω = M − 8K. According to [16, Theorem 2] the phase portrait of a systemfrom the family (S(h)


Vul5 ⇔ D < 0, Uc = 0;

Vul6 ⇔ D < 0, Uc 6= 0;

Ham12 ⇔ R = 0, U < 0;

Ham13 ⇔ R = U = 0, V 6= 0;

Ham19 ⇔

{D > 0, orD = 0, R 6= 0, Uc = 0;

Ham20 ⇔ R = 0, U > 0, W2 6= 0;

Ham21 ⇔ R = 0, U > 0, W2 = 0;

Ham22 ⇔ D = 0, Uc 6= 0.

(7.9)

For systems (S(h)3 ) calculations yield

D = 108c4(ab + 2β)2 − 4c2(b2 + 2ac− 4α)3,

Uc = −108c4(ab + 2β), R = 12c2x2,

µ0 = 0, M = −72x2, D = 48D, B1 = Uc/2,

µ1 = 4cx, L = 24x2 > 0, K = −4x2 < 0,

(7.10)

7.3.1. The case R 6= 0. In this case the condition c 6= 0 (i.e., µ1 6= 0) holds andas L > 0 and K < 0 according to [28, Table 4] at infinity the behavior of thetrajectories corresponds to Figure 9; i.e., we have the following configuration ofsingularities N,

(12

)PEP −H.

We observe that in the case D = 0 and (i.e., D = 0) by (7.9) we have the phaseportrait Ham19 with one finite singularity (if Uc = 0) and Ham22 with two finitesingularities (if Uc 6= 0). Considering the diagram (Figure 1), (7.10) and the fact,that the condition D2 + U2

c 6= 0 implies R 6= 0, we conclude that the conditionsfor determining the phase portraits from Tables 4 and 5 corresponding to the caseµ0 = 0 and µ1 6= 0 are equivalent to the respective conditions of (7.9) (i.e. theconditions for the phase portraits Vul5,Vul6, Ham19 and Ham22).

7.3.2. The case R = 0Then c = 0 and for systems (S(h)

3 ) we calculate

U = (b2 − 4α)[(ab + 2β)x− (b2 − 4α)y

]2x4, W2 = −6x4(ab + 2β),

µ0 = µ1 = 0, µ2 = (4α− b2)x2, U = U , B3 = 3W2,

κ = κ1 = 0, K2 = 768(b2 − 4α)x2.

(7.11)

We observe that sign(U) = sign(U) = − sign(µ2) = sign(K2).


Assume first U < 0. According to (7.9) the phase portrait of a system (S(h)3 )

corresponds to Ham12 (without real finite singularities). On the other hand asµ0 = µ1 = κ = κ1 = 0, µ2 > 0, L > 0 and K2 < 0, according to [28, Table 4] atinfinity the behavior of the trajectories corresponds to Figure 8, i.e. we arrive tothe configuration N,

(22

)H −H.

Admit now U > 0. Then by (7.9) we obtain the phase portrait Ham20 if W2 6= 0 andHam21 if W2 = 0 in both cases having two finite integrable saddles. As regardingthe configuration of infinite singularities we observe that µ2 < 0, L > 0 and K <0. So following [28, Table 4] we obtain Figure 9, i.e. we get the configurationN,

(22

)P E P − P E P . This leads to the total configuration 21 of Table 4. It

remains to note that by (7.11) the condition W2 = 0 is equivalent to B3 = 0.Assume finally U = 0. Then α = b2/4 and for systems (S(h)

3 ) we have

µ0 = µ1 = µ2 = µ3 = 0, µ4 = (ab + 2β)2x4/4 = V . (7.12)

According to (7.9) the phase portrait of a system (S(h)3 ) corresponds to Ham13 if

V 6= 0 and we get degenerate systems (with the phase portrait Ham14) if V = 0. Weclaim that in the first case at infinity we have the configuration N,

(42

)PHP−PHP .

Indeed following [28] for systems (S(h)3 ) we have

κ = κ1 = 0, L = 24x2 > 0, K = −4x2 < 0, R = −8x2 < 0

and according to [28, Table 4] at infinity the behavior of the trajectories correspondsto Figure 28; i.e., we arrive to the mentioned above configuration and our claim isproved.

It remains to observe that in the case V = 0 (then µ4 = 0) and the degeneratesystems have the phase portrait Ham14. Hence the singular invariant line coincideswith the invariant line of the respective linear systems, and using our notations (seepage 7) we get the configuration

( [|]; s

);N,

( [|];N

).

Considering that the condition M 6= 0 for the family (S(h)3 ) holds and (7.11), we

conclude that the conditions for determining the phase portraits from Tables 4 and5 corresponding to the case µ0 = µ1 = 0 are equivalent to the respective conditionsfrom (7.9) (i.e. the conditions for the phase portraits Ham12,Ham13, Ham20 andHam21).

7.4. The family of systems (S(h)4 ). For this family of systems we have η =

µ0 = µ1 = 0 and M = 0. According to [28, Table 2] at infinity we have one realsingularity of total multiplicity at least five. So in order to determine exactly theconfiguration of the singularities at infinity we shall use again Table 4 from [28].


According to [16, Theorem 1] the phase portrait of a system from the family(S(h)


Vul2 ⇔ P 6= 0, U > 0;

Ham11 ⇔

P 6= 0, U < 0, orP = U = 0, V 6= 0, orP = U = V = 0, W1 < 0;

Ham15 ⇔ P = U = V = 0, W1 > 0;

Ham16 ⇔ P = U = V = W1 = 0;

Ham17 ⇔ P 6= 0, U = 0;

Ham18 ⇔ P = 0, U 6= 0.

(7.13)

For systems (S(h)4 ) calculations yield

P = c4x4, U =[(ac− b2)2 + 4c(bα + cβ)

](bx + cy)2x4,

η = M = µ0 = µ1 = 0, µ2 = c2x2, K = 0, P = P , U = U .(7.14)

7.4.1. The case P 6= 0. In this case according to (7.13) a system from the family(S(h)

4 ) has one of the following phase portraits: Vul2 (if U > 0), Ham11 (if U < 0)or Ham17 (if U = 0). In all the cases at infinity we have a multiple node, whichaccording to [28, Table 4] is of multiplicity five, as the conditions µ0 = µ1 = M =K = 0 and µ2 > 0 (as c 6= 0) are verified. More precisely at infinity we have Figure

30, i.e. the nilpotent singular point(23

)N .

7.4.2. The case P = 0. Then c = 0 and for systems (S(h)4 ) we have

U = b6x6, V =[(α2 − abα− b2β)x− b2αy

]x3,

µ2 = 0, µ3 = b3x3, U = U , µ4 = V , K = 0, K1 = −bx3.(7.15)

So if U 6= 0 (i.e., b 6= 0) according to (7.13) we get the phase portrait Ham18 witha finite integrable saddle. On the other hand as µ3K1 = −b4x4 < 0 according to[28, Table 4] the configuration of infinite singularities corresponds to Figure 33; i.e.,(33

)PEPEP − P .Assume U = 0 (i.e., b = 0). In this case we have

µ3 = 0, µ4 = V = α2x4, W1 = (a2 + 4β)x4 − 4αx3y = B6, K3 = 0 (7.16)

and we shall consider two subcases: µ4 6= 0 and µ4 = 0.

(1) If µ4 6= 0 systems (S(h)4 ) are non-degenerate having the phase portrait Ham11

(see (7.13), as V 6= 0). Clearly at infinity we have a singular point of multiplicityseven. As µ4 > 0 and K3 = 0 according to [28, Table 4] at infinity we get Figure30. More exactly we have the configuration

(43

)P −P as the infinite singular point

is intricate.(2) Assume finally µ4 = 0, i.e. α = 0. Then systems (S(h)

4 ) with c = b = α = 0become degenerate possessing the phase portraits indicated respectively in (7.13).We observe that in the case of all the three phase portraits Ham11, Ham15 and


Ham16 the systems (S(h)4 ) have an invariant singular conic which is reducible. More

exactly, it splits into two parallel invariant lines which are real and distinct if B6 > 0(portrait Ham15); they are complex if B6 < 0 (portrait Ham11), and they coincide ifB6 = 0 (portrait Ham16). This leads to the respective configurations of singularitiesdescribed in Table 4 (see rows No. 28–30).

It remains to note taking into account (7.14), (7.15) and (7.16) that for the familyof systems (S(h)

4 ) the respective conditions for determining the phase portraits fromTables 4 and 5 in all the cases considered above are equivalent to the respectiveconditions from (7.13).

As all the cases are considered Theorem 1.2 is proved.

Acknowledgements. The first two authors are partially supported by grantsMTM 2008-03437 from MICINN/FEDER, and 2009SGR 410 from AGAUR. Thesecond author also is supported by ICREA Academia.

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Joan C. ArtesDepartament de Matematiques, Universitat Autonoma de Barcelona, 08193 Bellaterra,Barcelona, Spain

E-mail address: [email protected]

Jaume LlibreDepartament de Matematiques, Universitat Autonoma de Barcelona, 08193 Bellaterra,Barcelona, Spain


Nicolae VulpeInstitute of Mathematics and Computer Science, Academy of Science of Moldova, 5Academiei str, Chisinau, MD-2028, Moldova


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